a nutrition center sells two types of health foods to mountain-climbing teams, Trailblazer Mix and Frontier Mix. Each package of the Trailblazer Mix sells for $9.75. Each package of the Frontier Mix sells for $9.50. The center wishes to maximize its profits this month. However, this month it must operate its business under a few limitations. Specifically, the center only has available 60 pounds of corn cereal and 120 pounds of wheat cereal. The formula for Frontier Mix is 2 pounds of corn cereal and 3 pounds of wheat cereal. How many packages of each health food should the center manufacture in order to maximize the profits? Define the variables, define the objective function, constraint inequalities

To solve this problem, we need to define variables, establish the objective function, and set up the constraint inequalities.

Let's define the variables:
Let x represent the number of packages of Trailblazer Mix.
Let y represent the number of packages of Frontier Mix.

Objective function:
The objective of the nutrition center is to maximize profits.
The profit from each package of Trailblazer Mix is $9.75, and the profit from each package of Frontier Mix is $9.50.
Therefore, the objective function can be defined as:
Profit = 9.75x + 9.50y

Now, let's establish the constraint inequalities based on the available cereal quantities:
The Trailblazer Mix requires no corn cereal, and the Frontier Mix requires 2 pounds of corn cereal for every package.
The available corn cereal is 60 pounds in total.
Constraint 1: 2y ≤ 60

The Trailblazer Mix requires no wheat cereal, and the Frontier Mix requires 3 pounds of wheat cereal for every package.
The available wheat cereal is 120 pounds in total.
Constraint 2: 3y ≤ 120

Additionally, the number of packages of Trailblazer Mix and Frontier Mix cannot be negative.
Constraint 3: x ≥ 0
Constraint 4: y ≥ 0

So, we have the following set of constraint inequalities:
2y ≤ 60
3y ≤ 120
x ≥ 0
y ≥ 0

Now, we can solve this system of inequalities to determine the optimal number of packages of each health food that will maximize the profits.